Efter maden lægger hun karkluden ved vasken, så den kan tørre til i morgen.

Because Danish often puts a time expression first. Efter maden means after the meal / after eating, and it takes the first position in the sentence.

In main clauses, Danish normally follows the V2 rule: the finite verb must come in the second position. So when Efter maden comes first, the verb lægger must come next:

• Efter maden lægger hun ...

Not:

• Efter maden hun lægger ...

That word order is one of the most important patterns in Danish.

Mad means food, while maden means the food / the meal.

In Efter maden, Danish uses the definite form because it refers to a specific meal or the eating that has just happened. In natural English, we often translate it simply as after dinner, after the meal, or after eating.

So:

• efter mad = after food in a more general sense
• efter maden = after the meal / after the food just mentioned or understood

This is again because of the V2 rule in Danish main clauses.

When the sentence begins with something other than the subject, such as Efter maden, the verb comes before the subject:

• Efter maden lægger hun karkluden ...

If the subject came first, then you would get:

• Hun lægger karkluden ved vasken efter maden.

Both are grammatical, but the word order changes depending on what comes first.

Karklud means dishcloth, dishrag, or kitchen cloth.
The ending -en makes it definite:

• en karklud = a dishcloth
• karkluden = the dishcloth

Danish often uses the definite form where English might also use the, especially when the object is understood in the situation.

Lægge means to lay / put something down in a horizontal position, or more generally to place something somewhere.

That makes sense with a dishcloth, because it is something you lay down:

• Hun lægger karkluden ved vasken. = She puts/lays the dishcloth by the sink.

This contrasts with verbs like:

• stille = place something standing upright
• sætte = set/place something, often onto a surface or into a position

For a cloth, lægge is the natural choice.

Ved vasken means by the sink or next to the sink.

• ved = by, near, next to
• vasken = the sink

If you said i vasken, that would mean in the sink, which gives a different image. The sentence suggests she places the cloth near the sink so it can dry, not inside the sink.

So:

• ved vasken = by the sink
• i vasken = in the sink

Because the sentence refers to a specific sink, the one in the kitchen or the one already understood from the situation.

• en vask = a sink
• vasken = the sink

Danish often uses the definite form when the thing is obvious from context.

Here så means so that.

It introduces the purpose or intended result of the action:

• ... så den kan tørre til i morgen.
• ... so that it can dry by tomorrow.

This is different from så meaning then or so in other contexts. In this sentence, it connects the first action with its purpose.

Because after så in this sentence, Danish uses a subordinate clause.

In subordinate clauses, Danish does not use main-clause V2 word order. The subject usually comes before the finite verb:

• så den kan tørre

Not:

• så kan den tørre

So the contrast is:

• main clause: Efter maden lægger hun ...
• subordinate clause: så den kan tørre ...

This is a very common difference between Danish main and subordinate clauses.

Because karklud is a common-gender noun in Danish:

• en karklud
• therefore: den

If the noun were a neuter noun with et, you would use det instead.

So:

• en karklud → den
• et håndklæde → det

Literally, kan tørre means can dry.

Here kan does not mainly express permission. It expresses possibility or the idea that the cloth is being left there so it will be able to dry.

So the meaning is something like:

• so that it can dry
• so it can dry out
• so it has a chance to dry by tomorrow

Without kan, så den tørrer til i morgen would sound more like a direct statement that it dries by tomorrow, rather than the intended purpose.

It means dry by tomorrow or be dry by tomorrow.

The preposition til here is often used in time expressions with the sense of by the time of or for a certain point in time:

• til i morgen = by tomorrow / for tomorrow

So the sentence means she leaves the cloth there so it will be dry when tomorrow comes.

Yes, a version with for at is possible, but it is structured differently.

You could say:

• Efter maden lægger hun karkluden ved vasken for at lade den tørre til i morgen.

But så den kan tørre is very natural because it directly expresses purpose in an everyday way: she puts it there so that it can dry.

So:

• så den kan tørre = very natural, simple purpose/result wording
• for at ... = also possible, but often needs a slightly different construction

The present tense in Danish is often used for habitual actions or to describe a normal routine.

So this sentence can mean something like:

• After the meal, she puts the dishcloth by the sink so it can dry by tomorrow.

It sounds like a regular thing she does, not necessarily something happening only right now. English works the same way in many cases.